The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
Quantum chemistry models atoms and molecules using quantum mechanics, with the electronic structure problem at its core. This problem, which determines the energy and properties of electrons in a molecule, is exponentially hard to solve exactly on classical computers. To approximate solutions, effective models introduce layers of simplifications. However, these approximations can fail for strongly correlated systems, necessitating alternative approaches. Quantum computers, capable of processing quantum information natively, offer a promising path to simulate molecular systems beyond classical reach, with potential applications in catalysis, materials science, and photochemistry. Despite rapid advancements in quantum hardware, practical quantum computations remain challenging due to limited resources. This thesis optimizes quantum resource usage in computational chemistry, introducing novel embedding techniques and exploring applications beyond ground-state energies. Chapter 1 establishes the foundation by reviewing quantum chemistry methods, from Hartree-Fock and Density Functional Theory (DFT) to active space approaches. It also provides an introduction to quantum computing for chemistry, focusing on near-term algorithms. The following chapters present novel methods aimed at reducing quantum resource requirements and tackling problems classical methods struggle with. Chapter 2 introduces a technique to reduce the λ-norm of the electronic structure Hamiltonian, a key measure of quantum resource demand. By transforming molecular orbitals, this method enhances sampling efficiency in Noisy Intermediate-Scale Quantum (NISQ) algorithms and reduces runtime in Fault-Tolerant Quantum Computing (FTQC). Localized orbitals outperform canonical ones, and an orbital-optimization scheme is proposed to minimize the λ-norm. Benchmarks on hydrogen chains, FeMoco, and Ruthenium complexes demonstrate its effectiveness. Chapter 3 presents the FragPT2 framework for simulating large systems with multiple interacting fragments. This method embeds each fragment in the mean field of others and applies multi-reference perturbation theory to capture inter-fragment correlations such as dispersion and charge transfer efficiently. The approach reduces computational overhead and enables scalable simulations, as shown in applications to \ce{N2} dimers, aromatic dimers, and butadiene. Chapter 4 focuses on photochemical systems, where multiple electronic states must be treated equally. The state-averaged orbital-optimized variational quantum eigensolver (SAOOVQE) is extended with analytical energy gradients and non-adiabatic couplings, enabling geometry optimizations and simulations of non-adiabatic dynamics on quantum hardware. This is demonstrated by locating the conical intersection in formaldimine, highlighting SAOOVQE’s potential for studying photoisomerization in biomolecules. Building on this, Chapter 5 develops a quantum algorithm to detect conical intersections, where the Born-Oppenheimer approximation breaks down. These intersections are characterized by a discrete Berry phase (0 or π), representing their topological nature. The algorithm extracts this phase as a single bit of information, ensuring resilience to noise. Analytical proof and numerical validation using a toy molecular model illustrate its effectiveness for studying light-perception biochemistry and related photochemical processes. Finally, Chapter 6 explores machine learning density functionals trained on noisy quantum data. While DFT is computationally efficient, its accuracy depends on approximated exchange-correlation functionals. This work proposes a neural network-based approach to learn the exact functional from quantum data, showing that training on noisy outputs from NISQ devices can mitigate sampling errors. Applications to the Fermi-Hubbard model and other systems demonstrate its potential to generalize and provide accurate results under noise. This thesis advances quantum computational chemistry by optimizing quantum resource usage, developing new embedding strategies, and expanding the scope of quantum simulations beyond ground-state properties. These contributions pave the way for practical quantum algorithms that could revolutionize molecular simulations in the near future.
The advancements in classical computational components such as CPUs and GPUs have had a reverberating impact on the democratization of computational power available, with the field of computational chemistry reaping significant benefits from the propulsion of the limits of systems studied. The accurate prediction of chemical properties and reaction dynamics through computational simulation is now a centerpiece of modern chemical research, significantly accelerating discoveries research areas of materials science and drug development, as well as industrially relevant processes such as the emission of aldehydes from soft foams, flame retardants and pyrolysis and the carbonylation of epoxides. These achievements of computational chemistry still are overshadowed in the face of fundamental bottlenecks where classical methods remain constrained by the exponential scaling when treating large and/or strongly correlated quantum many-body systems. This innate computational complexity often constitutes chemically accurate calculations based on first principles impractical, or plain impossible, for systems beyond modest sizes. Quantum computing offers a potential pathway around these limitations, originally formally formulated by David Deutsch as a tool for simulating quantum systems, with the first quantum algorithms indicating speed-ups compared to classical counterparts arising from Peter Shor and Lov Grover for the prime factorization of an integer number and unstructured search respectively. Relying on fundamental theorems by Claude Shannon, it has been demonstrated that through the use of error-correcting codes fault-tolerant quantum computers are able to safeguard information against noise and decoherence, strongly indicating towards exponential speedups in solving central computational chemistry problems, such as determining ground state energies or reaction pathways of chemical systems. Substantial strides in experimental quantum computing over the past decade, including proof-of-principle demonstrations of quantum error correction, mean to bring full-scale fault-tolerant quantum computing within the grasp of humanity. The breakthroughs needed to attain scalable, error-corrected quantum computation with capabilities exceeding those of classical methods in real-world scenarios is likely still some years away. While striving to address the engineering challenges standing in the way of achieving fault tolerant quantum computing, substantial effort has been focused toward developing and benchmarking hybrid algorithms tailored for the Noisy Intermediate- Scale Quantum (NISQ) devices, available now to researchers. These hybrid quantum-classical algorithms, with the most notable being the Variational Quantum Eigensolver (VQE), allow the probing of quantum computing’s utility in computational chemistry even in the face of significant hardware noise and limited coherence times. The work presented in this thesis introduces several key additions the VQE framework, geared especially towards to NISQ hardware, where the mindful use of available resources is imperative. First, the implementation of the Conjugate Model Search optimizer is described, an optimization method that integrates simple models that are surrogate to the original function, combined with conjugate gradient optimization to navigate effectively parameter landscapes typical of variational algorithms, prone to noise. Second, this thesis contains the proposal and extensive study of a hyperparameter tuning procedure employing genetic algorithms, allowing for comprehensive simulation benchmarks that augment the robustness and efficiency of VQE implementations with respect to convergence time through real wall-time models of quantum processors. Continuing, a bridge is presented between quantum simulation and quantum chemistry, through qubit-to-fermion mappings of existing quantum gates, further enabling the direct and accurate modeling of chemical systems on trapped atom quantum hardware with the added arsenal provided by techniques designed for qubit architectures. Additionally, experimental data contained in this thesis include detailed benchmarking data collected from state-of-the-art trapped ion quantum processors available at the Johannes Gutenberg Universität Mainz, demonstrating the actual viability and performance of various methodologies for extracting results. Lastly, the state of VQE is examined, as it served as a springboard for a plethora of advancements in the development of quantum computing hardware, as well as the direction the field of simulation of computational quantum chemistry via quantum devices is headed towards in the coming years.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
The practical application of quantum technologies to chemical problems faces significant challenges, particularly in the treatment of realistic basis sets and the accurate inclusion of electron correlation effects. A direct approach to these problems is currently infeasible due to limitations in the number of logical qubits, their fidelity, and the shallow circuit depths supported by existing hardware, all of which hinder simulations at the required level of accuracy. A promising alternative is hybrid quantum-classical computing, where classical resources are used to construct effective Hamiltonians characterized by dimensions that conform to the constraints of current quantum devices. In this paper, we demonstrate the performance of a hybrid approach: coupled cluster downfolded Hamiltonians are first evaluated in reduced-dimensionality active spaces, and the corresponding ground-state energies are subsequently computed using quantum algorithms. Our comprehensive analysis explores the achievable accuracy in recovering correlation energies when hundreds of orbitals are downfolded into a problem size tractable by today’s quantum hardware. We argue that such flexible hybrid algorithms, where problem size can be tailored to available quantum resources, can serve as a bridge between noisy intermediate-scale quantum devices and future fault-tolerant quantum computers, marking a step toward the early realization of quantum advantage in chemistry.
Quantum mechanical problems are among the hardest to simulate and, in some cases, remain intractable even for the most powerful computers. Quantum computing has emerged as a new technological platform to address such challenges, with rapid advances in recent years. Yet, current quantum devices remain noisy and limited in scale. Hence, it is essential to identify classically hard problems of practical interest and tractable with existing quantum devices. Among potential applications, the real-time simulation of quantum systems is one of the most promising to deliver an early, practical quantum advantage. This doctoral thesis is therefore centered around simulating quantum dynamics on quantum devices. We first present an overview of the most relevant quantum algorithms for quantum dynamics, highlighting respective advantages and limitations. Further, we identify relevant problems within quantum dynamics that could benefit from quantum simulation in the near future. Second, we propose a method for benchmarking hardware and error mitigation algorithms that is based on well-understood theoretical results and suffers from no scaling issues. The resulting quality metric is intuitive and transferable to other applications. We successfully implement the scheme on up to 133 qubits, demonstrating coherent evolution up to a two-qubit gate depth of 28, featuring 1396 two-qubit gates, before noise becomes prevalent. Third, we propose a novel variant of probabilistic error amplification to implement open quantum dynamics, relying on characterizing and altering hardware noise to mimic the system-environment interaction under study. Lastly, we present two studies on state preparation and phase classification: first, a hybrid algorithm to prepare ground states of electron-phonon systems; second, a quantum machine learning-based approach to distinguish phases in previously prepared quantum states.
The study of quantum many-body systems is crucial to our understanding of nature at the microscopic scale. As analytical solutions quickly become intractable, the development of efficient computational simulation techniques is indispensable to predict the behaviour of quantum systems. This task is particularly challenging as the dimension of the underlying Hilbert space grows exponentially with the system size. While classical computational methods have achieved remarkable success in specific regimes, their applicability is fundamentally limited by this exponential scaling. Quantum computers offer a natural platform for representing and simulating quantum systems, but current devices are constrained by noise and limited qubit counts, drastically restricting the algorithms that can be run. In this context, hybrid quantum-classical algorithms provide a promising approach to combine the strengths of both paradigms. This thesis develops hybrid quantum-classical algorithms for the simulation of quantum many-body systems. The key idea is to perform classically difficult tasks on near-term quantum devices, while leveraging classical computational power to control, correct, and extend the obtained results. Within this framework, we introduce two complementary approaches for the simulation of quantum dynamics. The first method employs circuit knitting techniques to distribute variational quantum simulations across multiple quantum devices. The entanglement between weakly coupled clusters of highly correlated spins is recovered in classical post-processing, where the overhead is controlled through a constrained optimisation problem. The second method combines a simplified, Trotterised quantum dynamics simulation with a parametrised classical ansatz trained to correct for the approximations made on the quantum device. We show that this method can be employed to correct Trotter errors, implement long-range gates with hardware-efficient circuits, or simulate larger systems by incorporating additional degrees of freedom purely classically. In addition, we present a hybrid approach to computing thermal properties of quantum many-body systems via the Quantum Finite Temperature Lanczos Method. A Krylov subspace is generated on a quantum computer by combining the efficient preparation of typical states with real-time evolution. A classical computer then diagonalises the system in the low-dimensional subspace and estimates the finite-temperature expectation values through trace estimation. The results presented in this thesis demonstrate that hybrid quantum-classical algorithms can expand the range of systems that can be studied on near-term quantum devices. Conversely, we show that incorporating quantum subroutines into classical simulation techniques can improve their efficiency. These results suggest that, rather than treating classical and quantum simulation as competing paradigms, their careful integration can provide a more effective route toward practical quantum many-body simulation.